Energy Dissipation in Synchronous Binary Asteroids

2025-05-06 0 0 2.73MB 24 页 10玖币

侵权投诉

Alex J. Meyera,∗,Daniel J. Scheeresa,Harrison F Agrusab,Guillaume Noisetc,Jay McMahona,

Özgür Karatekinc,Masatoshi Hirabayashid,e and Ryota Nakanod

aSmead Department of Aerospace Engineering, University of Colorado, Boulder, CO 80303, USA

bDepartment of Astronomy, University of Maryland, College Park, MD 20742, USA

cRoyal Observatory of Belgium, 3 Avenue Circulaire, 1180 Brussels, Belgium

dDepartment of Aerospace Engineering, Auburn University, Auburn, AL 36849, USA

eDepartment of Geosciences, Auburn University, Auburn, AL 36849, USA

ARTICLE INFO

Keywords:

Asteroids, dynamics

Satellites of asteroids

Tides, solid body

Near-Earth objects

ABSTRACT

Synchronous binary asteroids can experience libration about their tidally-locked equilibrium, which

will result in energy dissipation. This is an important topic to the Asteroid Impact and Deﬂection

Assessment, where excitation caused by the DART kinetic impact in the Didymos binary asteroid

system may be reduced through dissipation before Hera arrives to survey the eﬀects of the impact.

We develop a numeric model for energy dissipation in binary asteroids to explore how diﬀerent sys-

tem conﬁgurations aﬀect the rate of energy dissipation. We ﬁnd tumbling within the synchronous

state eliminates a systematic trend in libration damping on short timescales (several years), but not

over long times (hundreds of years) depending on the material conditions. Furthermore, damping of

libration, eccentricity, and ﬂuctuations in the semimajor axis are primarily dependent on the stiﬀness

of the secondary, whereas the semimajor axis secular expansion rate is dictated by the stiﬀness of

the primary, as expected. Systems experiencing stable planar libration in the secondary can see a

noticeable reduction in libration amplitude after only a few years depending on the stiﬀness of the

secondary, and thus dissipation should be considered during Hera’s survey of Didymos. For a very

dissipative secondary undergoing stable libration, Hera may be able to calculate the rate of libration

damping in Dimorphos and therefore constrain its tidal parameters.

1. Introduction

The Asteroid Impact and Deﬂection Assessment (AIDA)

is a collaboration supported by NASA and ESA to test the

feasibility of a kinetic impactor to deﬂect a small asteroid for

the purpose of planetary defense (Cheng et al.,2018). Two

missions will combine results to produce the most accurate

knowledge possible on the ﬁrst kinetic impact of an asteroid:

NASA’s DART (Double Asteroid Redirection Test), which

will perform the actual kinetic impact (Rivkin et al.,2021),

and ESA’s Hera, which will assess the eﬀectiveness of the

impact several years later (Michel et al.,2022). The target

of the impact is Dimorphos, the secondary in the Didymos

binary asteroid system. By impacting Dimorphos, DART

will change the mutual orbit period around Didymos, and

ground-based measurements of the orbit period change will

reveal how much momentum was transferred to Dimorphos.

Approximately 4 years after the DART impact, Hera is sched-

uled to rendezvous with the Didymos system to perform a

detailed analysis of the post-impact system, making several

key measurements.

The degree to which the system’s dynamics will evolve

through energy dissipation between the DART impact and

Hera’s arrival remains an open question for AIDA. While

this window is only around 4 years, a rubble-pile structure –

like Didymos is hypothesized to be based on earlier dynam-

∗Corresponding author at 3775 Discovery Dr, Boulder, CO 80303,

USA

E-mail address: alex.meyer@colorado.edu

ORCID(s): 0000-0001-8437-1076 (A.J. Meyer); 0000-0002-3544-298X

(H.F. Agrusa); 0000-0002-1649-7176 (G. Noiset); 0000-0002-1821-5689 (M.

Hirabayashi); 0000-0002-9840-2416 (R. Nakano)

ics and geological studies (Agrusa et al.,2022;Walsh,2018;

Walsh et al.,2008;Jacobson and Scheeres,2011a) – may

be very eﬃcient at dissipating energy, and thus this prob-

lem warrants attention. The question of energy dissipation

after the DART impact and prior to Hera’s rendezvous with

Didymos is necessary in order to maximize the scientiﬁc and

practical return of the AIDA collaboration. As Hera charac-

terizes the spin state of Dimorphos, it is important to under-

stand how the current spin state has changed since the impact

in order to fully comprehend the eﬀects of the DART impact.

Ignoring dissipation in the system may lead to an incorrect

estimation of the eﬃcacy of a kinetic impactor during Hera’s

survey. While the scientiﬁc implications of this work extend

beyond AIDA and Didymos to binary asteroid dynamics in

general, we focus our analysis to this application given its

current relevance and the wealth of analysis on Didymos and

the DART impact in the literature.

While binary asteroids provide an ideal test site for plan-

etary defense missions given their short mutual orbit peri-

ods (Cheng et al.,2018), they also oﬀer a chance to study

the unique dynamics of the full 2-body problem (F2BP).

Given the asteroids’ close proximity and generally asym-

metric shapes (Pravec et al.,2016), their orbital motion is

strongly coupled with their attitude, leading to complex dy-

namics (Maciejewski,1995;Scheeres,2006,2009). Through

this strong coupling, the bodies’ spins and mutual orbit will

evolve concurrently while energy dissipation occurs. Addi-

tionally, spin-orbit coupling can lead to attitude instabilities

as a result of orbit perturbations such as the DART impact

(Agrusa et al.,2021).

There are two main mechanisms of energy dissipation

A. J. Meyer et al.: Preprint submitted to Elsevier Page 1 of 24

arXiv:2210.10877v1 [astro-ph.EP] 19 Oct 2022

Energy Dissipation in Synchronous Binary Asteroids

we will consider in this work, both stemming from the de-

formation of the bodies. The ﬁrst is tidal torque, in which

the tidal forces of both bodies act to move the system into a

synchronous equilibrium (Murray and Dermott,1999;Gol-

dreich and Sari,2009;Taylor and Margot,2010). The sec-

ond is non-principal axis (NPA) rotation, in which rotation

about any axis other than the major principal axis will dis-

sipate energy until the major principal axis is aligned with

the angular momentum (Burns et al.,1973;Breiter et al.,

2012;Molina et al.,2003;Ershkov and Leshchenko,2019;

Pravec et al.,2005). Both these mechanisms will drive the

system toward a conﬁguration in which the two asteroids are

mutually tidally locked, with their spin angular momentum

vectors aligned with their major principal axes and the orbit

angular momentum vector (Taylor and Margot,2011). We

call this state the doubly-synchronous equilibrium.

While many studies focus on energy dissipation in the

two-body problem, they generally ignore the speciﬁc dynam-

ical regime that Didymos will inhabit after the DART im-

pact: a system which is generally synchronous but with nonzero

libration of the secondary (Taylor and Margot,2010;Goldre-

ich and Sari,2009). Here we deﬁne libration as any angular

deviation of the secondary’s long axis away from the tidally

locked conﬁguration, but smaller than 90◦so the secondary

remains on-average synchronous. Generally there are two

modes of libration: free and forced (Murray and Dermott,

1999). While forced libration is driven by eccentricity, free

libration is governed by the average libration over an or-

bit period and is thus eccentricity-agnostic (Tiscareno et al.,

2009). Given the strongly coupled nature of binary asteroids,

we make no distinction between these two modes and sim-

ply adopt the physical libration angle. This study will focus

exclusively on this dynamic state and so also carries scien-

tiﬁc merit beyond the speciﬁc application of the AIDA col-

laboration. More recently, Efroimsky (2018) analyzed en-

ergy dissipation in a tidally perturbed librating body. This

is the same regime we are interested in here, but we attempt

to relax the small-libration assumption from that work and

extend results to binary asteroids, which orbit much closer

than planet-moon systems. Another noteworthy study is that

of Jacobson and Scheeres (2011a), who apply a tidal torque

model to binary asteroids. However, this analysis is limited

to 2 dimensions, whereas we are interested in the full 3 di-

mensional dynamics. Quillen et al. (2020,2022) study tidal

dissipation in coupled systems with some attention spent on

the libration state, and our work falls in a similar vein but we

focus on how diﬀerent shapes and stiﬀness of the secondary

aﬀect the dissipation process.

Since the main motivation of this study is the AIDA col-

laboration, we ﬁrst provide background on Didymos, the DART

impact, and previous analyses on the post-impact dynamics

in Section 2. We then derive our dynamical model, includ-

ing dissipation mechanisms, in Section 3. Results on energy

dissipation are presented in Section 4, and we validate these

results by comparing with a higher-ﬁdelity numeric model

in Section 5. The implications for Hera over the short-term

are investigated in Section 6. In Section 7we discuss the

possible implications of the BYORP eﬀect, and in Section 8

we investigate how the dissipation behavior depends on the

material parameters. Finally, we present a discussion and

our conclusions in Section 9.

2. Background

We will apply our dissipation model to the Didymos sys-

tem, which we nominally assume is in a singly-synchronous

equilibrium prior to any perturbation, with the secondary’s

rotation period equal to the orbit period. The rationale for

this assumption is outlined in Richardson et al. (2022). To

calculate this equilibrium we adopt the method described in

Agrusa et al. (2021) and iterate the system bulk density un-

til the stroboscopic orbit period matches the observed value.

We deﬁne the stroboscopic orbit period as the time required

for the secondary to traverse 360◦relative to an inertial ob-

server, akin to a lightcurve observation. This approach means

we have developed our own independent estimate of the sys-

tem density rather than using values derived from observa-

tions, although our density lies within the error bars of the

observed value (Naidu et al.,2020;Scheirich and Pravec,

2022). We calculate the stroboscopic orbit period using the

method outlined in Meyer et al. (2021). The resulting equi-

librium system has the parameters outlined in Table 1. We

will assume a triaxial shape for the secondary, but note chang-

ing the axis ratios of Dimorphos does not appreciably aﬀect

the equilibrium parameters of the system. While keeping the

mean radius of Dimorphos constant, we will vary its axis ra-

tios (𝑎∕𝑏and 𝑏∕𝑐, with 𝑎 > 𝑏 > 𝑐) to investigate how the

shape of Dimorphos aﬀects the energy dissipation rate.

In this work we will focus on two shapes of the sec-

ondary, one with 𝑎∕𝑏= 1.2,𝑏∕𝑐= 1.1, and the other with

𝑎∕𝑏= 1.4,𝑏∕𝑐= 1.3. In conjunction with the mean radius,

we can solve for the semiaxes that deﬁne the ellipsoid, as

well as the dimensionless shape parameter 𝑆deﬁned as

𝑆=𝐵−𝐴

𝐶(1)

where 𝐴,𝐵, and 𝐶are the three principal moments of in-

ertia of the ellipsoid corresponding to the axes 𝑎,𝑏, and 𝑐,

respectively. Table 2gives the dimensions of the two ellip-

soids we will primarily use as Dimorphos, as well as their

shape parameter 𝑆.

The DART impact will push Dimorphos out of the equi-

librium state (Meyer et al.,2021;Agrusa et al.,2021). The

impact can be quantiﬁed by the momentum enhancement

factor known as 𝛽, which is deﬁned as the ratio of the true

system momentum change to the momentum carried by the

impactor. Mathematically, this is described as

𝛽=𝑝𝑡𝑟𝑢𝑒

𝑝𝑖𝑚𝑝𝑎𝑐𝑡𝑜𝑟

.(2)

𝛽can be converted into a change in velocity using the rela-

tionship

Δ⃗𝑣 =𝑀𝑖𝑚𝑝𝑎𝑐𝑡𝑜𝑟

𝑀𝐵⃗𝑢 + (𝛽− 1) ̂𝑢 ⋅⃗𝑢̂𝑛(3)

A. J. Meyer et al.: Preprint submitted to Elsevier Page 2 of 24

Energy Dissipation in Synchronous Binary Asteroids

Table 1

Summary of the equilibrium Didymos system prior to the DART impact, from Pravec

et al. (2022), Scheirich and Pravec (2022), Naidu et al. (2020), and Scheirich and Pravec

(2009). Our density estimate diﬀers from that reported in Scheirich and Pravec (2022) as

we calculate it using a dynamical approach, but our solution falls within the 1𝜎derived

error bars.

Parameter Symbol Value Notes

Orbit Period 𝑃𝑜𝑟𝑏 11.92 hr Measured (Pravec et al.,2022;Scheirich and Pravec,2022)

Didymos Rotation Period 𝑃𝐴2.26 hr Measured (Pravec et al.,2022)

Didymos Mean Radius 𝑅𝐴390 m Derived (Naidu et al.,2020)

Dimorphos Mean Radius 𝑅𝐵82 m Derived (Naidu et al.,2020;Scheirich and Pravec,2009)

System Bulk Density 𝜌2.2 g/cm3Derived here, similar to (Scheirich and Pravec,2022)

Semimajor Axis 𝑎1200 m Measured (Naidu et al.,2020)

Eccentricity 𝑒0 Assumed (Scheirich and Pravec,2022;Richardson et al.,2022)

Inclination 𝑖0◦Assumed (Scheirich and Pravec,2022;Richardson et al.,2022)

Table 2

Summary of the two triaxial ellipsoids used throughout this

work as the secondary shape.

Shape 𝑎[m] 𝑏[m] 𝑐[m] 𝑆

𝑎∕𝑏= 1.2,𝑏∕𝑐= 1.195.6 79.7 72.4 0.18

𝑎∕𝑏= 1.4,𝑏∕𝑐= 1.3112 80.0 61.5 0.32

where 𝑀𝑖𝑚𝑝𝑎𝑐𝑡𝑜𝑟 is the impactor mass, 𝑀𝐵is the mass of

Dimorphos, ⃗𝑢 is the impactor velocity, and ̂𝑛 is the outward

surface normal at the impact site (Rivkin et al.,2021;Feld-

hacker et al.,2017). For this analysis we will assume ̂𝑛 is par-

allel to the velocity of Dimorphos, and that ⃗𝑢 is misaligned

with the velocity vector by 10 degrees out of the orbit plane

and 10 degrees in the radial direction, consistent with Richard-

son et al. (2022). Note the impact is retrograde, decreas-

ing the mutual orbit period while increasing its eccentricity.

From Richardson et al. (2022), we use an impactor mass of

536 kg and velocity of 6.143 km/s relative to the secondary.

For this analysis, we will assume a perturbation equiva-

lent to a 𝛽value of 3, as this is large enough to excite unstable

motion in some shapes of Dimorphos, but small enough to

allow stable motion in other shapes, and also lies in the ex-

pected range of 𝛽:1< 𝛽 ≲ 6(Raducan and Jutzi,2022;

Stickle et al.,2022). A perturbation of 𝛽= 3 is roughly

equivalent to increasing the eccentricity to around 0.02, de-

pending on the secondary shape and mass. This allows us to

study both stable and unstable dynamical regimes without

having to test multiple perturbation magnitudes. Note this

selection is not grounded in the actual DART impact, as we

are interested in how diﬀerent secondary shapes aﬀect dis-

sipation rates within a system rather than making any quan-

titative predictions, as an accurate prediction is impossible

without knowledge of the system’s interior structure. Fol-

lowing the DART impact and Hera survey, this analysis can

be revisited with better constraints on the shape and mass of

Dimorphos. We reproduce the results of Agrusa et al. (2021)

for 𝛽= 3 in Fig. 1to show the unstable region of motion.

While the size of Dimorphos is ﬁxed by the bulk diameter,

the shape of the triaxial ellipsoid is deﬁned by the axis ratios

𝑎∕𝑏and 𝑏∕𝑐, where 𝑎,𝑏, and 𝑐are the longest, intermediate,

and shortest semiaxes of the ellipsoid, respectively. Fig. 1

shows the amplitude of the 1-2-3 Euler angles, correspond-

ing to roll, pitch, and yaw, for each secondary shape. If a

system is in a true equilibrium, these angles would remain

zero. An unstable region is apparent in Fig. 1where some

secondary shapes result in tumbling. The unstable region in

which the secondary begins to tumble is outlined by a yellow

dashed line. This is not a formal boundary for this region and

is only intended to aid interpretation. The unstable region is

dependent on the system’s eccentricity, so this boundary can

not be applied outside of our impact scenario.

Due to the spin-orbit coupling in binary asteroids, these

systems are non-Keplerian, and thus osculating Keplerian

elements can be somewhat misleading. In an equilibrium

conﬁguration, the secondary may appear to be in a circular

orbit to an external observer, but the Keplerian orbit is el-

liptical. In this conﬁguration, the secondary is trapped at

periapsis while the orbit itself precesses. Thus, there is a

non-zero eccentricity at equilibrium and the semimajor axis

is not the same as the separation distance (Scheeres,2009).

However, these elements are still useful as they can give us

an idea of the system’s secular evolution over time, and we

use the Keplerian osculating elements throughout this work.

3. Dynamical Model

The mutual dynamics of binary asteroids are character-

ized by the F2BP, in which the orbit and attitude of the bodies

are coupled. This leads to complex dynamics, and various

models have been developed to simulate these systems with

varying trade-oﬀs between ﬁdelity and computational cost.

Since we are concerned with timespans of many years, it is

necessary to select a more basic model at the cost of reduced

ﬁdelity. In this context, we are more concerned with the sys-

tem’s qualitative behavior over long time periods rather than

short-term accuracy, so this is a ﬁne compromise. As such,

we model Didymos as a spherical primary and Dimorphos as

A. J. Meyer et al.: Preprint submitted to Elsevier Page 3 of 24

Energy Dissipation in Synchronous Binary Asteroids

Figure 1: The maximum amplitude of the 1-2-3 Euler angles for an impact corresponding

to 𝛽= 3 (𝑒≈ 0.02), from the simulation set from Agrusa et al. (2021). The unstable

regions, indicated by nonzero amplitudes in the roll and pitch angles, are governed by

intersections of various resonances among fundamental frequencies of the system. The

unstable region is outlined by the yellow dashed line; this is not a formal boundary and

only serves to aid interpretation.

an ellipsoidal secondary, which allows for full 3D dynamics

with an elongated secondary without becoming too compu-

tationally expensive. We will validate this model against a

high-ﬁdelity model in Section 5. Fig. 2shows a diagram of

the system, where body 𝐴is Didymos and body 𝐵is Dimor-

phos. In later discussions, quantities with subscripts 𝐴or 𝐵

specify those for the designated bodies.

Figure 2: Diagram showing the dynamic model.

The equations of motion are straightforward and we omit

any derivation, instead referring the reader to Scheeres (2006).

These equations are deﬁned in the body-ﬁxed frame of the

secondary. We have six degrees of freedom, being the rela-

tive separation and the rotation of Dimorphos. The equations

of motions are

⃗𝑟+2 ⃗𝜔𝐵×̇

⃗𝑟+̇

⃗𝜔𝐵×⃗𝑟+⃗𝜔𝐵×( ⃗𝜔𝐵×⃗𝑟) = (𝑀𝐴+𝑀𝐵)𝜕𝑈

𝜕⃗𝑟 (4)

𝐈𝐵⋅̇

⃗𝜔𝐵+⃗𝜔𝐵×𝐈𝐵⋅⃗𝜔𝐵= −𝑀𝐴𝑀𝐵⃗𝑟 ×𝜕𝑈

𝜕⃗𝑟 (5)

where 𝐈𝐵is the inertia tensor of the secondary, which is a

simple diagonal matrix in the secondary’s body-ﬁxed frame.

𝑈is the gravitational potential around the ellipsoidal Dimor-

phos, and to ease computation time we use a second degree

expansion in the form of MacCullagh’s formula (Murray and

Dermott,1999):

𝑈= − 𝑀𝐴𝑀𝐵

𝑟−𝑀𝐴(𝐴+𝐵+𝐶− 3Φ)

2𝑟3(6)

where 𝐴,𝐵, and 𝐶are respectively the minimum, interme-

diate, and maximum principal inertia values of Dimorphos,

and Φis a quantity deﬁned by

Φ = 𝐴𝑥2+𝐵𝑦2+𝐶𝑧2

𝑟2(7)

with (𝑥, 𝑦, 𝑧)being the Cartesian coordinates of the primary

in the secondary’s body-ﬁxed frame so that ⃗𝑟 =𝑥̂

𝑖+𝑦̂

𝑗+𝑧̂

𝑘.

We next need to introduce the methods of dissipation

through non-rigid processes, both through tidal torque and

NPA rotation, which rise from a combination of deforma-

tion, rotation, and translation of the bodies (Hirabayashi,2022).

We ignore any surface motion on both the primary and sec-

ondary, which includes rotation-induced granular motion on

the secondary’s surface and any associated body reshaping,

which changes the gravitational potential energy (Agrusa et al.,

2022;Agrusa et al.,2022), tidal saltation and YORP-induced

landslides on the primary (Harris et al.,2009), and boul-

der movement on either body (Brack and McMahon,2019),

which would also dissipate energy. We will assume any re-

shaping and surface motion to be small and intermittent, and

the energy dissipated by these events to be negligible over

time. So our estimates on damping times for Didymos can

be considered conservative for a given set of material proper-

ties since additional mechanisms will only increase the dis-

sipation rate.

3.1. Tidal Torque

To describe energy dissipation from the system, we in-

troduce equations for tidal torque to add to our dynamic model.

In selecting a basic model for tidal torque, we have two choices:

the constant 𝑄model, in which the rate of dissipation is

driven by the ratio of tidal quality factor 𝑄and the simple

A. J. Meyer et al.: Preprint submitted to Elsevier Page 4 of 24

Energy Dissipation in Synchronous Binary Asteroids

Love number 𝑘2(Murray and Dermott,1999), or the con-

stant time lag model, in which the angle between the tidal

bulge and the line connecting the two bodies is a constant

Δ𝑡(Mignard,1979;Hut,1981). Based on the physics of

our problem setup, the secondary will librate about the syn-

chronous conﬁguration, and thus a constant lag angle would

be inappropriate, as the lag angle should oscillate as a re-

sult of the libration. For this reason we select the constant

𝑄model, which is the same model adopted by Jacobson and

Scheeres (2011a), in which the tidal torque is deﬁned as:

Γ=−sign(𝜔−𝜔𝑜𝑟𝑏)3

23

4𝜋𝜌 2𝐺𝑀2

𝐴𝑀2

𝐵

𝑟6𝑅

𝑘

𝑄(8)

where the body’s angular velocity is 𝜔, the orbit’s angular

rate is 𝜔𝑜𝑟𝑏,𝑅is the reference radius for the body, 𝜌is its den-

sity, and 𝑄∕𝑘is the tidal dissipation ratio. The tidal quality

factor 𝑄is related to the tidal bulge lag angle (𝑄∼ 1∕ sin 𝜖),

while the love number 𝑘2describes the level of body defor-

mation due to the tidal potential. Henceforth we drop the

subscript 2 on the Love number for simplicity. A large value

of 𝑄∕𝑘corresponds to a more stiﬀ body that dissipates more

slowly. In reality, the tidal dissipation is far more compli-

cated than simply selecting constant values for the unknown

𝑄∕𝑘values. As pointed out by Efroimsky (2015), tidal dis-

sipation in binary asteroids, including rubble piles, may be

governed primarily by the body’s viscosity, rather than rigid-

ity. Others, including Goldreich and Sari (2009) and Nimmo

and Matsuyama (2019), argue that friction is a critical pa-

rameter. Since there is no current estimate for the viscosity

of rubble pile asteroids to the authors’ knowledge, we adopt

the friction approach. Furthermore, while many studies as-

sume the quality number 𝑄to be constant, this parameter

depends on the tidal frequency. Further complicating this re-

lationship is the fact that the tidal quality number is not a lin-

ear function of the tidal frequency, and can either increase or

decrease with the frequency (Ferraz-Mello,2013). We also

are left with the problem of the tidal lag angle oscillation.

To address this we adopt the same solution as Jacobson and

Scheeres (2011a); we will linearize the tidal torque around

the point where (𝜔−𝜔𝑜𝑟𝑏)is near zero so the torque does

not immediately switch signs as the secondary librates (see

Appendix C therein for details on this linearization). This

linearization is necessary, as the tidal bulge is a physical phe-

nomenon and requires a ﬁnite time to cross between leading

or trailing the tide-raising body.

Note that Taylor and Margot (2010) point out the simple

tidal model assumes the two bodies are widely separated,

whereas the separation between Didymos and Dimorphos

is only slightly larger than 3 primary radii. In their work,

Taylor and Margot (2010) calculate that higher order terms

in the tidal potential speed up the process of tidal evolu-

tion. However, they also ﬁnd that uncertainties in the sys-

tem, particularly surrounding 𝑄∕𝑘, dominate over the higher

order tidal expansion. Thus, we continue with the simple

tidal model given the large uncertainty associated with the

bulk system density and physical properties, while keeping

in mind higher order terms in the tidal model will only in-

crease the rate of damping in the system. Thus, the error as-

sociated with this tidal model is considered to be secondary

to the considerable uncertainty on the 𝑄∕𝑘coeﬃcient for

our purposes.

Unfortunately, given the lack of knowledge on the phys-

ical parameters of rubble piles, particularly their viscosity,

we cannot calculate an accurate value for 𝑄∕𝑘. For lack of a

better option, we surrender ourselves to the typical simpliﬁ-

cations surrounding the factor 𝑄∕𝑘, and we turn to the work

by Nimmo and Matsuyama (2019), who derive an estimate

for a constant 𝑄∕𝑘for rubble pile binary asteroids, which

can be approximated by

𝑄

𝑘≈ 300𝑅(9)

for 𝑅in meters. This leads to a value for the primary 𝑄𝐴∕𝑘𝐴≈

1 × 105and for the secondary 𝑄𝐵∕𝑘𝐵≈ 2.5 × 104. Note this

expression for 𝑄is frequency dependent and derived for a

non-synchronous binary system. However, we again empha-

size there is large uncertainty associated with these values so

this deﬁnition serves as a ﬁrst-order approximation, as the

error from uncertainties dominates over the error from the

assumptions. Furthermore, these values are consistent with

existing estimates for small bodies in the literature (Brasser,

2020;Jacobson and Scheeres,2011b;Scheirich et al.,2015,

2021). However, there is not a consensus on this linear scal-

ing. For example, Goldreich and Sari (2009) propose an in-

verse scaling of 𝑄∕𝑘with 𝑅, and even in their own work

Nimmo and Matsuyama (2019) point out a scaling with 𝑅3∕2

may be more accurate. Another consideration is if Dimor-

phos turns out to be monolithic instead of a rubble pile, its

𝑄∕𝑘value would likely be orders of magnitude higher (Gol-

dreich and Sari,2009). Hence, the large uncertainty in 𝑄∕𝑘

dominates over other errors associated with our model, and

it is futile to develop a high ﬁdelity tidal model while lim-

ited by this unknown parameter. Given the large uncertainty

and lack of consensus around 𝑄∕𝑘, we adopt the linear scal-

ing only as nominal parameters, and subsequently investi-

gate how varying 𝑄∕𝑘for both the primary and secondary

aﬀects the system behavior later in Section 8.

Returning to the tidal torque equation, this model is still

only deﬁned in 2-dimensions and we wish to extend this to

a full 3-dimensional analysis, as out-of-plane rotation of the

secondary is a possibility. This can be done with only a few

corrections to the classic model. To start, we need to de-

ﬁne a vector for the torque direction. The torque will act

to push the spin rates of the asteroids into the synchronous

equilibrium, but physically cannot act in the direction of the

position vector of the secondary relative to the primary. We

deﬁne the relative spin rate of a body:

⃗

𝜙=⃗𝜔 −⃗𝜔𝑜𝑟𝑏.(10)

We can then use this spin vector as the vector along which

the torque acts, with a small correction so the torque in the

radial direction is zero:

Γ=−

⃗

𝜙− ( ̇

⃗

𝜙⋅̂𝑟)̂𝑟

̇

⃗

𝜙− ( ̇

⃗

𝜙⋅̂𝑟)̂𝑟

(11)

A. J. Meyer et al.: Preprint submitted to Elsevier Page 5 of 24

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

EnergyDissipationinSynchronousBinaryAsteroidsAlexJ.Meyera,<,DanielJ.Scheeresa,HarrisonFAgrusab,GuillaumeNoisetc,JayMcMahona,ÖzgürKaratekinc,MasatoshiHirabayashid,eandRyotaNakanodaSmeadDepartmentofAerospaceEngineering,UniversityofColorado,Boulder,CO80303,USAbDepartmentofAstronomy,UniversityofMaryland...

展开>> 收起<<

Energy Dissipation in Synchronous Binary Asteroids.pdf

共24页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Energy Dissipation in Synchronous Binary Asteroids

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: